Problem: Simplify and expand the following expression: $ \dfrac{2k}{2k - 1}+\dfrac{k}{5k + 9} $
Explanation: In order to add expressions, they must have a common denominator. Get both fractions over a common denominator of $(2k - 1)(5k + 9)$ Multiply the first term by $\dfrac{5k + 9}{5k + 9}$ $ \begin{align*} \dfrac{2k}{2k - 1} \times \dfrac{5k + 9}{5k + 9} & = \dfrac{(2k)(5k + 9)}{(2k - 1)(5k + 9)} \\ & = \dfrac{10k^2 + 18k}{(2k - 1)(5k + 9)}\end{align*} $ Multiply the second term by $\dfrac{2k - 1}{2k - 1}$ $ \begin{align*} \dfrac{k}{5k + 9} \times \dfrac{2k - 1}{2k - 1} & = \dfrac{(k)(2k - 1)}{(5k + 9)(2k - 1)} \\ & = \dfrac{2k^2 - k}{(5k + 9)(2k - 1)}\end{align*} $ Now we have: $ = \dfrac{10k^2 + 18k}{(2k - 1)(5k + 9)} + \dfrac{2k^2 - k}{(5k + 9)(2k - 1)} $ Now both terms have a common denominator we can simply add the numerators: $ = \dfrac{10k^2 + 18k + 2k^2 - k}{(2k - 1)(5k + 9)} $ $ = \dfrac{12k^2 + 17k}{(2k - 1)(5k + 9)}$ Expand the denominator: $ = \dfrac{12k^2 + 17k}{10k^2 + 13k - 9}$